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Conformal Transformations

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The technique is based on transforming geometries using functions of the complex ... The analysis also uses the conformal transformation method: Neil Marks; DLS/CCLRC ... – PowerPoint PPT presentation

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Title: Conformal Transformations


1
Conformal Transformations
  • Neil Marks,
  • DLS/CCLRC,
  • Daresbury Laboratory,
  • Warrington WA4 4AD,
  • U.K.
  • Tel (44) (0)1925 603191
  • Fax (44) (0)1925 603192

2
Use of transformations
  • This is a mathematical technique developed to
    give analytical expressions for potential
    and flux density distributions for (simple)
    geometries.
  • This was the standard technique for magnet design
    before finite element analysis (f.e.a.) codes
    were developed.
  • The technique is based on transforming
    geometries using functions of the complex
    variable.
  • We start with a model with unknown field and
    potential distributions and transform that to a
    geometry where distributions can be analytically
    defined. The known distribution is then
    transformed back to the initial model using the
    inverse of the first transform, giving the
    required result.
  • Each transformation often involves an
    intermediate step.

3
The Transformations.
  • We start by defining two complex planes, Z and W

All conformal transformations preserve the
angle of intersection between two curves
except at the origin and at poles of the
transforming function. As the most suitable
ideal pole to use for the known distribution
will be a pair of parallel poles extending
to plus and minus infinity, such a
function will be necessary.
4
The 'Schwarz - Christoffel' Transformation
  • This is one type of a conformal transform it
    transforms polygons (complete or open) in the
    Z plane to a straight line on the real
    axis in the W plane
  • The transformation is given by
  • dZ/dW M ( W - a1)((a1/p ) -1) (W - a2)((a2/p )
    - 1) ........
  • where M is an arbitrary constant to be determined.

5
Second transformation.
  • A second transformation is then used to
    translate from a T plane, where the
    geometry is an infinite dipole, back into
    the W plane
  • The two transformations are then combined
    to predict the distributions in the real
    magnet.

6
Example a dipole end.

Transformation Z to W
dZ/dW M (W 1)1/2 ( W ) -1 Z M 2(W
1)1/2 ln (W 1)1/2 -1 - ln (W 1)1/2 1
N where N is another arbitrary constant N
0 just sets the origin in Z plane for W
-1, the above expression gives Z i M p i
g/2 so M g/2p for W gt 0, Z is real.
7
Transformation T to W
  • The T plane, an infinite dipole
  • dT/dW M (W a )(a/p 1)
  • for the above a 0 a 0
  • so dT/dW M W -1
  • T M ln(W) N
  • again N 0 sets the origin in the Z plane
  • for W -1, above expression gives Z Mip
    ig/2
  • so M g/2p, giving
  • T (g/2p) ln W
  • W exp(2pT/g)

8
Resulting equation
  • We now substitute for W to give Z in terms of T
  • Z (g/2p) 2 exp (2pT/g) 11/2
  • ln exp(2pT/g 1 -1
  • - ln exp(2pT/g 1 1
  • Equipotential lines are Im (T) const
  • Flux lines are Re (T) const.
  • Expanding with Re and Im of T and then separating
    the real and imaginary components of Z is long
    and detailed see next slide.

9
Solution

For g 1
where
10
Graphical results - lines of scalar potential

11
Graphical results - lines of flux

12
Flux density B
  • B -? f
  • put T y if
  • then Bx - ?f / ? x
  • By - ? f / ? y
  • Bx - j By - ? f/? x i ? f/? y
  • from Cauchy-Riemann equations
  • ? y / ? x ? f / ? y
  • ? y / ? y - ? f/? x
  • so Bx - iBy
  • - ? f / ? x i? y / ? x
  • i ? T / ? x
  • hence
  • B d T/ dZ
  • ( dT/dW )( dW/dZ )

Flux density on x axis
13
Problems
  • integration is only analytical for angles of 0
    or multiples of p/2
  • and a limited number of right angles
  • other more complex geometries require numerical
    integration
  • predicts distributions only for µ ? in the
    steel
  • the technique takes no account of coils ie all
    currents are at infinity.

14
The 'Rogowski' roll-off
  • The classical end solution, developed originally
    in electrostatics during the study of the end
    effect for two parallel capacitor plates. The
    analysis also uses the conformal transformation
    method

15
Analysis
  • dZ/dW M (W 1)/W
  • Z (d/?)( 1 W ln W )
  • T (1/?) ln W
  • so Z (d/?)( 1 exp ?T ?T )
  • if T y jf
  • then in the Z plane
  • x (d/?)(1 (exp py)(cos ?f) ?y)
  • y (d/?)( (exp ?y)(sin ?f) ?f)
  • Potential is lines of const f stream lines are
    const y .

16
Graphical results
  • Potential lines

17
Blown-up version

The central heavy line is for f 0.5. Rogowski
showed that this was the fastest changing line
along which the field intensity was monotonically
decreasing.
18
Application to magnet ends
  • Conclusion Recall that a high µ steel surface is
    a line of constant scalar potential. Hence, a
    magnet pole end using the f 0.5 potential line
    will see a monotonically decreasing flux density
    normal to the steel at some point where B is
    much lower, this can break to a vertical end
    line.

Magnet half gap height g/2 Centre line of
gap is y 0 Equation is y g/2
(g/?) exp ((?x/g)-1)
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